How could Deep Blue possibly cheat?
The allegation was based on the fact that Deep Blue made a choice that did not yield the immediate (or short term) benefit that was synonymous with systems back then (1997). Computational capability was significantly less powerful then, and Kasparov claimed that only a grand master would have made the decision that the system did - so the deep blue team cheated by having a human perform the move instead of the system.

He asked for a rematch, but IBM did not allow this, which only added to the suspicion.

This is a great article with deep analysis on the specific moves and circumstances - however suffice to say that Kasparov was trying to bait the system into making a decision for a weak pawn, but the system chose otherwise and instead put Kasparov into a compromised position:
https://en.chessbase.com/post/deep-blue-s-cheating-move